part of material not showing up in render
Found the solution:
I have a bevel modifier on the ring itself. I did apply it to a vertex group (the edges of the ring). I now changed it to "limit method: weight" after I gave those edges a mean weight of 1. Now it works fine.
The reason I didn't see the problem before in the viewport was that I accidentally had the modifier hidden in the viewport.
Shoutout to @Robert_Gützow for the right idea.